Should the prediction of the body temperature given a camera image be modelled as classification or regression?
I think it depends on you application and what data you have available. 

If the prediction of body temperature itself doesn't have to be accurate and classes like COLD, NORMAL, and HOT will suffice, you should stay with a classification. There isn't a cut off but as you increase the number of classes that represent numbers on the same scale, it may become more difficult to interpret the result as there will be a distribution across the classes.

If you choose regression on the other hand, you are not restricted by your classes anymore and may be able to tell the difference between 36.5 and 36C which (according to wikipedia) can be the difference between normal and cold. This is something classes may not be able to capture. 

Another thing to consider is what your training data looks like and how accurately you want to predict the temperature. If you have pictures of people and a temperature reading, where was the reading taken and how accurate is the reading? If it isn't accurate (+- 1 degree) you may not be able to give as accurate predictions as you would like and may only be able to do 3 different classes like above. 

If you don't have a data set, then that is another problem altogether and might require another question as it depends on your application. 

I think that your problem is interesting and I hope this can help you to understand how best to apply deep learning to it.